Finite Group Databasebeta

← #14426210

Subgroups — #14426210

Order 1920.

992 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#532C211yesmaximal, characteristic, complemented
2C2×#91839C212yesmaximal, complemented
3C4×#6817C211yesmaximal, characteristic, complemented
4#11751334C212yesmaximal, complemented
5#11751191C211yesmaximal, characteristic, complemented
6#11751184C211yesmaximal, characteristic, complemented
7#11751176C211yesmaximal, characteristic, complemented
8#11751173C211yesmaximal, characteristic, complemented
9#11751158C212yesmaximal, complemented
10#11751155C212yesmaximal, complemented
11C2×C4⋊(C3⋊#168)C211yesmaximal, characteristic, complemented
12#10615027—31nomaximal, Hall
13C2×C3×C5×#41C2212yes—
14C2×#6817C2211yescharacteristic
15C3×C5×C4⋊Q8C2211yescharacteristic
16C3×C5×#127C2211yescharacteristic
17C3×C5×#124C2212yes—
18C3×C5×#121—21no—
19C3×C5×#121—21no—
20C3×C4×C5×D4—21no—
21C3×C4×C5×D4—21no—
22C2×C3×C42×C5C2211yescharacteristic
23#91839—28no—
24C2×#6718C2212yes—
25C2×#6717C2212yes—
26C2×#6716C2212yes—
27C2×#6715C2212yes—
28C4×#1220—21no—
29C4×#1220—21no—
30#91815C2211yescharacteristic
31C22×C3⋊#168C2212yes—
32C4×D60C2211yescharacteristic
33C4×D60C2211yescharacteristic
34#91761—21no—
35C42×D15—21no—
36C4×Dic30C2211yescharacteristic
37C4×Dic30C2211yescharacteristic
38#91119C2212yes—
39#91119C2212yes—
40#91116C2212yes—
41#91116C2212yes—
42C4⋊(C3⋊#168)C2211yescharacteristic
43C4⋊(C3⋊#168)C2211yescharacteristic
44C4⋊(C3⋊#168)C2212yes—
45#83296—51nomaximal, Hall
46C5×#532D311yescharacteristic, complemented
47C2×#4096—32no—
48C4×#709—31no—
49#65522—32no—
50#65379—31no—
51#65372—31no—
52#65364—31no—
53#65361—31no—
54#65346—32no—
55#65343—32no—
56C2×C4⋊#168—31no—
57C3×C5×#41—24no—
58C3×C5×#41—24no—
59C2×C3×C5×Q8C2312yes—
60C2×C3×C5×D4—22no—
61C2×C3×C5×D4—21no—
62C2×C3×C5×D4—21no—
63C2×C3×C5×D4C2312yes—
64C22×C3×C4×C5—22no—
65C22×C3×C4×C5D411yescharacteristic
66C22×C3×C4×C5D411yescharacteristic
67C22×C3×C4×C5C2311yescharacteristic
68C2×#1220—21no—
69#6817—22no—
70#6817D414yescomplemented
71C2×D60C2311yescharacteristic
72C2×C4×D15—21no—
73C2×C4×D15—21no—
74C2×C4×D15—21no—
75C2×Dic30C2311yescharacteristic
76C3×C5×C4⋊C4—21no—
77C3×C5×C4⋊C4—21no—
78C3×C5×C22⋊C4—22no—
79C3×C5×C22⋊C4—22no—
80C3×C42×C5D411yescharacteristic
81C3×C42×C5D411yescharacteristic
82C3×C42×C5C2311yescharacteristic
83C3×C42×C5C2311yescharacteristic
84#6719—21no—
85#6718—24no—
86#6717—24no—
87#6716—24no—
88#6715—24no—
89#6714—21no—
90#6714—21no—
91C4⋊Dic15C2311yescharacteristic
92C4⋊Dic15C2311yescharacteristic
93#6712—21no—
94#6712—21no—
95C4×Dic15—21no—
96C4×Dic15—21no—
97C4×Dic15—21no—
98C2×C3⋊#168—24no—
99C2×C3⋊#168C2312yes—
100C2×C3⋊#168C2312yes—
101C3×#532D511yescharacteristic, complemented
102C2×#950—52no—
103C4×#235—51no—
104#5233—52no—
105#5090—51no—
106#5083—51no—
107#5075—51no—
108#5072—51no—
109#5057—52no—
110#5054—52no—
111C2×C4⋊(C3⋊C8)—51no—
112C2×C5×#41C2×D312yes—
113C2×#709—31no—
114C5×C4⋊Q8C2×D311yescharacteristic
115C5×#127C2×D311yescharacteristic
116C5×#124C2×D312yes—
117C5×#121—21no—
118C5×#121—21no—
119C4×C5×D4—21no—
120C4×C5×D4—21no—
121C2×C42×C5C2×D311yescharacteristic
122#4096—68no—
123C2×#689—32no—
124C2×#688—32no—
125C2×#687—32no—
126C2×#686—32no—
127C4×C22⋊D5—61no—
128C4×C22⋊D5—61no—
129#4072—31no—
130C22×#168—32no—
131C4×D20—31no—
132C4×D20—31no—
133#4018—61no—
134C42×D5—61no—
135C4×Dic10—31no—
136C4×Dic10—31no—
137#3967—32no—
138#3967—32no—
139#3964—32no—
140#3964—32no—
141C4⋊#168—32no—
142C4⋊#168—31no—
143C4⋊#168—31no—
144#3035—151noSylow
145C23×C3×C5—22no—
146C23×C3×C5C2×D411yescharacteristic
147C22×D15—21no—
148C3×C5×Q8—22no—
149C3×C5×Q8—22no—
150C3×C5×D4—22no—
151C3×C5×D4—22no—
152C3×C5×D4—42no—
153C3×C5×D4—44no—
154C3×C5×D4—42no—
155C2×C3×C4×C5—22no—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5—41no—
158C2×C3×C4×C5—44no—
159C2×C3×C4×C5C2×D411yescharacteristic
160C2×C3×C4×C5C2×D411yescharacteristic
161C2×C3×C4×C5C2×D411yescharacteristic
162C2×C3×C4×C5C2×D411yescharacteristic
163C2×C3×C4×C5C2×D412yes—
164C2×C3×C4×C5C2×D411yescharacteristic
165C2×C3×C4×C5#4112yes—
166C2×C3×C4×C5C2411yescharacteristic
167#1220—24no—
168C2×Dic15—21no—
169C2×Dic15—21no—
170C2×Dic15—21no—
171D60—21no—
172D60C2×D412yes—
173C4×D15—24no—
174C4×D15—42no—
175C4×D15—42no—
176Dic30—21no—
177Dic30C2×D412yes—
178C3⋊#168—22no—
179C3⋊#168—22no—
180C2×C3×#41C2×D512yes—
181C2×#235—51no—
182C3×C4⋊Q8C2×D511yescharacteristic
183C3×#127C2×D511yescharacteristic
184C3×#124C2×D512yes—
185C3×#121—21no—
186C3×#121—21no—
187C3×C4×D4—21no—
188C3×C4×D4—21no—
189C2×C3×C42C2×D511yescharacteristic
190#950—108no—
191C2×#216—52no—
192C2×#215—52no—
193C2×#214—52no—
194C2×#213—52no—
195C4×#67—101no—
196C4×#67—101no—
197#926—51no—
198C22×C3⋊C8—52no—
199C4×D12—51no—
200C4×D12—51no—
201#872—101no—
202C42×D3—101no—
203C4×Dic6—51no—
204C4×Dic6—51no—
205#835—52no—
206#835—52no—
207#832—52no—
208#832—52no—
209C4⋊(C3⋊C8)—52no—
210C4⋊(C3⋊C8)—51no—
211C4⋊(C3⋊C8)—51no—
212C5×#41—24no—
213C5×#41—24no—
214C2×C5×Q8C22×D312yes—
215C2×C5×D4—22no—
216C2×C5×D4—21no—
217C2×C5×D4—21no—
218C2×C5×D4C22×D312yes—
219C22×C4×C5—22no—
220C22×C4×C5#6711yescharacteristic
221C22×C4×C5#6711yescharacteristic
222C22×C4×C5C22×D311yescharacteristic
223C2×C22⋊D5—61no—
224#709—34no—
225#709—62no—
226C2×D20—31no—
227C2×C4×D5—61no—
228C2×C4×D5—61no—
229C2×C4×D5—61no—
230C2×Dic10—31no—
231C5×C4⋊C4—21no—
232C5×C4⋊C4—21no—
233C5×C22⋊C4—22no—
234C5×C22⋊C4—22no—
235C42×C5#6711yescharacteristic
236C42×C5#6711yescharacteristic
237C42×C5C22×D311yescharacteristic
238C42×C5C22×D311yescharacteristic
239C22⋊Dic5—61no—
240#689—64no—
241#688—64no—
242#687—64no—
243#686—64no—
244#685—61no—
245#685—61no—
246C4⋊Dic5—31no—
247C4⋊Dic5—31no—
248#683—61no—
249#683—61no—
250C4×Dic5—61no—
251C4×Dic5—61no—
252C4×Dic5—61no—
253C2×#168—32no—
254C2×#168—32no—
255C2×#168—64no—
256C2×#135—152no—
257#532D1511yescharacteristic, complemented
258C4×#41—151no—
259#464—152no—
260#463—152no—
261#462—151no—
262#461—151no—
263#460—151no—
264#459—151no—
265C2×C4⋊C8—151no—
266#416—152no—
267C22×C3×C5—21no—
268C22×C3×C5—21no—
269C22×C3×C5—42no—
270C22×C3×C5—42no—
271C22×C3×C5—42no—
272C22×C3×C5#12112yes—
273C22×C3×C5C22×D411yescharacteristic
274C2×D15—22no—
275C2×D15—41no—
276C2×D15—41no—
277C3×C4×C5—21no—
278C3×C4×C5—21no—
279C3×C4×C5—21no—
280C3×C4×C5—42no—
281C3×C4×C5#12112yes—
282C3×C4×C5#13512yescomplemented
283C3×C4×C5C22×D411yescharacteristic
284C3×C4×C5C2×#4111yescharacteristic
285Dic15—22no—
286Dic15—41no—
287Dic15—41no—
288C3×#41—24no—
289C3×#41—24no—
290C2×C3×Q8C22×D512yes—
291C2×C3×D4—22no—
292C2×C3×D4—21no—
293C2×C3×D4—21no—
294C2×C3×D4C22×D512yes—
295C22×C3×C4—22no—
296C22×C3×C4C22⋊D511yescharacteristic
297C22×C3×C4C22⋊D511yescharacteristic
298C22×C3×C4C22×D511yescharacteristic
299C2×#67—101no—
300#235—54no—
301#235—102no—
302C2×D12—51no—
303C2×C4×D3—101no—
304C2×C4×D3—101no—
305C2×C4×D3—101no—
306C2×Dic6—51no—
307C3×C4⋊C4—21no—
308C3×C4⋊C4—21no—
309C3×C22⋊C4—22no—
310C3×C22⋊C4—22no—
311C3×C42C22⋊D511yescharacteristic
312C3×C42C22⋊D511yescharacteristic
313C3×C42C22×D511yescharacteristic
314C3×C42C22×D511yescharacteristic
315#217—101no—
316#216—104no—
317#215—104no—
318#214—104no—
319#213—104no—
320(C2×D3)⋊C4—101no—
321(C2×D3)⋊C4—101no—
322C4⋊Dic3—51no—
323C4⋊Dic3—51no—
324Dic3⋊C4—101no—
325Dic3⋊C4—101no—
326C4×Dic3—101no—
327C4×Dic3—101no—
328C4×Dic3—101no—
329C2×C3⋊C8—52no—
330C2×C3⋊C8—52no—
331C2×C3⋊C8—104no—
332C23×C5—22no—
333C23×C5C2×#6711yescharacteristic
334C22×D5—61no—
335C5×Q8—22no—
336C5×Q8—22no—
337C5×D4—22no—
338C5×D4—22no—
339C5×D4—42no—
340C5×D4—44no—
341C5×D4—42no—
342C2×C4×C5—22no—
343C2×C4×C5—22no—
344C2×C4×C5—41no—
345C2×C4×C5—44no—
346C2×C4×C5D3×D412yes—
347C2×C4×C5#23712yes—
348C2×C4×C5C2×#6711yescharacteristic
349C2×C4×C5C2×#6711yescharacteristic
350C2×C4×C5C2×#6711yescharacteristic
351C2×C4×C5C2×#6711yescharacteristic
352C2×C4×C5C2×#6711yescharacteristic
353C2×C4×C5C23×D311yescharacteristic
354C22⋊D5—64no—
355C2×Dic5—61no—
356C2×Dic5—61no—
357C2×Dic5—61no—
358D20—32no—
359D20—61no—
360C4×D5—64no—
361C4×D5—122no—
362C4×D5—122no—
363Dic10—32no—
364Dic10—61no—
365#168—62no—
366#168—62no—
367C2×#41—151no—
368C2×#41C2×D1512yes—
369#135—308no—
370C2×Q16—152no—
371C2×SD16—152no—
372C2×SD16—152no—
373C2×D8—152no—
374C22×C8—152no—
375C4⋊Q8C2×D1511yescharacteristic
376#127C2×D1511yescharacteristic
377#124C2×D1512yes—
378#121—21no—
379#121—21no—
380C4×Q8—151no—
381C4×Q8—151no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—151no—
385C4×D4—151no—
386C4×D4—301no—
387C4×D4—301no—
388#117—151no—
389#117—301no—
390C2×C42—301no—
391C2×C42C2×D1511yescharacteristic
392C4⋊C8—151no—
393C4⋊C8—151no—
394C4⋊C8—152no—
395#103—152no—
396#103—152no—
397#102—152no—
398#102—152no—
399C2×C3×C5—22no—
400C2×C3×C5—82no—
401C2×C3×C5C2×#12111yescharacteristic
402C2×C3×C5C2×#13511yescharacteristic
403C2×C3×C5#57511yescharacteristic
404D15—42no—
405C23×C3—22no—
406C23×C3C2×C22⋊D511yescharacteristic
407C22×D3—101no—
408C3×Q8—22no—
409C3×Q8—22no—
410C3×D4—22no—
411C3×D4—22no—
412C3×D4—42no—
413C3×D4—44no—
414C3×D4—42no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—41no—
418C2×C3×C4—44no—
419C2×C3×C4D4×D512yes—
420C2×C3×C4#71112yes—
421C2×C3×C4C2×C22⋊D511yescharacteristic
422C2×C3×C4C2×C22⋊D511yescharacteristic
423C2×C3×C4C2×C22⋊D511yescharacteristic
424C2×C3×C4C2×C22⋊D511yescharacteristic
425C2×C3×C4C2×C22⋊D511yescharacteristic
426C2×C3×C4C23×D511yescharacteristic
427#67—104no—
428C2×Dic3—101no—
429C2×Dic3—101no—
430C2×Dic3—101no—
431D12—52no—
432D12—101no—
433C4×D3—104no—
434C4×D3—202no—
435C4×D3—202no—
436Dic6—52no—
437Dic6—101no—
438C3⋊C8—102no—
439C3⋊C8—102no—
440C22×C5—21no—
441C22×C5—21no—
442C22×C5—42no—
443C22×C5—42no—
444C22×C5—42no—
445C22×C5#93812yes—
446C22×C5C22×#6711yescharacteristic
447C2×D5—62no—
448C2×D5—121no—
449C2×D5—121no—
450C4×C5—21no—
451C4×C5—21no—
452C4×C5—21no—
453C4×C5—42no—
454C4×C5#93812yes—
455C4×C5#95012yescomplemented
456C4×C5C2×D3×D411yescharacteristic
457C4×C5C2×#23711yescharacteristic
458Dic5—62no—
459Dic5—121no—
460Dic5—121no—
461#41—24no—
462#41—24no—
463#41—154no—
464#41—302no—
465C2×Q8—151no—
466C2×Q8C22×D1512yes—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—151no—
471C2×D4—301no—
472C2×D4C22×D1512yes—
473C22×C4—22no—
474C22×C4—301no—
475C22×C4—301no—
476C22×C4—301no—
477C22×C4#122011yescharacteristic
478C22×C4#122011yescharacteristic
479C22×C4C22×D1511yescharacteristic
480Q16—304no—
481SD16—304no—
482SD16—304no—
483D8—304no—
484C2×C8—152no—
485C2×C8—152no—
486C2×C8—304no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—151no—
490C4⋊C4—151no—
491C4⋊C4—301no—
492C4⋊C4—301no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—301no—
496C22⋊C4—301no—
497C22⋊C4—301no—
498C42—301no—
499C42—301no—
500C42—301no—
501C42#122011yescharacteristic
502C42#122011yescharacteristic
503C42C22×D1511yescharacteristic
504C42C22×D1511yescharacteristic
505C3×C5#303511yescharacteristic, complemented, Hall
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3#408412yes—
512C22×C3C22×C22⋊D511yescharacteristic
513C2×D3—102no—
514C2×D3—201no—
515C2×D3—201no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—42no—
520C3×C4#408412yes—
521C3×C4#409612yescomplemented
522C3×C4C2×D4×D511yescharacteristic
523C3×C4C2×#71111yescharacteristic
524Dic3—102no—
525Dic3—201no—
526Dic3—201no—
527C2×C5—22no—
528C2×C5—82no—
529C2×C5C2×#93811yescharacteristic
530C2×C5C2×#95011yescharacteristic
531C2×C5#582211yescharacteristic
532D5—122no—
533C23—22no—
534C23—301no—
535C23C2×#122011yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—152no—
539Q8—301no—
540D4—22no—
541D4—22no—
542D4—42no—
543D4—44no—
544D4—42no—
545D4—152no—
546D4—304no—
547D4—301no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—301no—
553C2×C4—304no—
554C2×C4—301no—
555C2×C4—301no—
556C2×C4—602no—
557C2×C4—602no—
558C2×C4D4×D1512yes—
559C2×C4#681912yes—
560C2×C4C2×#122011yescharacteristic
561C2×C4C2×#122011yescharacteristic
562C2×C4C2×#122011yescharacteristic
563C2×C4C2×#122011yescharacteristic
564C2×C4C2×#122011yescharacteristic
565C2×C4C23×D1511yescharacteristic
566C8—302no—
567C8—302no—
568C2×C3—22no—
569C2×C3—82no—
570C2×C3C2×#408411yescharacteristic
571C2×C3C2×#409611yescharacteristic
572C2×C3#6615611yescharacteristic
573D3—202no—
574C5#8329611yescharacteristic, complemented, Sylow
575C22—21no—
576C22—21no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—302no—
581C22—601no—
582C22—601no—
583C22#9182712yes—
584C22C22×#122011yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—42no—
589C4—302no—
590C4—601no—
591C4—601no—
592C4#9182712yes—
593C4#9183912yescomplemented
594C4C2×D4×D1511yescharacteristic
595C4C2×#681911yescharacteristic
596C3#1061502711yescharacteristic, complemented, Sylow
597C2—22no—
598C2—82no—
599C2—602no—
600C2C2×#9182711yescharacteristic
601C2C2×#9183911yescharacteristic
602C2#1175658111yescharacteristic
subgroupG/H
centerC2×C4C2×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#532C2
socleC22×C3×C5C22×D4
hypercenter#532D15
hypocenterC3×C5#3035
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×C5×#532C2semidirect
2. C2×#91839C2semidirect
3. C4×#6817C2semidirect
4. #11751334C2semidirect
5. #11751191C2semidirect
6. #11751184C2semidirect
7. #11751176C2semidirect
8. #11751173C2semidirect
9. #11751158C2semidirect
10. #11751155C2semidirect
11. C2×C4⋊(C3⋊#168)C2semidirect
46. C5×#532D3semidirect
70. #6817D4semidirect
101. C3×#532D5semidirect
257. #532D15semidirect
282. C3×C4×C5#135semidirect
455. C4×C5#950semidirect
505. C3×C5#3035semidirect
521. C3×C4#4096semidirect
574. C5#83296semidirect
593. C4#91839semidirect
596. C3#10615027semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C5×#5322
C2×#918392
C4×#68172
#117513342
#117511912
#117511842
#117511762
#117511732
#117511582
#117511552
C2×C4⋊(C3⋊#168)2
#106150273
#832965